↑

Properties

Label 2-90e2-1.1-c1-0-18
Degree 22
Conductor 81008100
Sign 11
Analytic cond. 64.678864.6788
Root an. cond. 8.042318.04231
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.28·7-s − 4.14·11-s + 6.52·13-s + 5.98·17-s + 7.17·19-s − 7.53·23-s − 5.19·29-s + 5.17·31-s + 5.24·37-s − 0.680·41-s − 1.28·43-s + 5.31·47-s − 5.35·49-s + 2.21·53-s + 7.60·59-s − 2.17·61-s − 15.6·67-s − 5.50·71-s − 7.80·73-s + 5.31·77-s + 6·79-s − 9.75·83-s + 10.0·89-s − 8.35·91-s + 14.3·97-s + 0.680·101-s − 2.56·103-s + ⋯
L(s)  = 1  − 0.484·7-s − 1.24·11-s + 1.80·13-s + 1.45·17-s + 1.64·19-s − 1.57·23-s − 0.964·29-s + 0.930·31-s + 0.861·37-s − 0.106·41-s − 0.195·43-s + 0.774·47-s − 0.765·49-s + 0.304·53-s + 0.990·59-s − 0.278·61-s − 1.90·67-s − 0.653·71-s − 0.913·73-s + 0.605·77-s + 0.675·79-s − 1.07·83-s + 1.06·89-s − 0.876·91-s + 1.45·97-s + 0.0677·101-s − 0.252·103-s + ⋯

Functional equation

Λ(s)=(8100s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8100s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 81008100    =    22⋅34⋅522^{2} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 64.678864.6788
Root analytic conductor: 8.042318.04231
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8100, ( :1/2), 1)(2,\ 8100,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9882888501.988288850
L(12)L(\frac12) ≈\approx 1.9882888501.988288850
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
5 1 1
good7 1+1.28T+7T2 1 + 1.28T + 7T^{2}
11 1+4.14T+11T2 1 + 4.14T + 11T^{2}
13 1−6.52T+13T2 1 - 6.52T + 13T^{2}
17 1−5.98T+17T2 1 - 5.98T + 17T^{2}
19 1−7.17T+19T2 1 - 7.17T + 19T^{2}
23 1+7.53T+23T2 1 + 7.53T + 23T^{2}
29 1+5.19T+29T2 1 + 5.19T + 29T^{2}
31 1−5.17T+31T2 1 - 5.17T + 31T^{2}
37 1−5.24T+37T2 1 - 5.24T + 37T^{2}
41 1+0.680T+41T2 1 + 0.680T + 41T^{2}
43 1+1.28T+43T2 1 + 1.28T + 43T^{2}
47 1−5.31T+47T2 1 - 5.31T + 47T^{2}
53 1−2.21T+53T2 1 - 2.21T + 53T^{2}
59 1−7.60T+59T2 1 - 7.60T + 59T^{2}
61 1+2.17T+61T2 1 + 2.17T + 61T^{2}
67 1+15.6T+67T2 1 + 15.6T + 67T^{2}
71 1+5.50T+71T2 1 + 5.50T + 71T^{2}
73 1+7.80T+73T2 1 + 7.80T + 73T^{2}
79 1−6T+79T2 1 - 6T + 79T^{2}
83 1+9.75T+83T2 1 + 9.75T + 83T^{2}
89 1−10.0T+89T2 1 - 10.0T + 89T^{2}
97 1−14.3T+97T2 1 - 14.3T + 97T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.76715879741981512004143848019, −7.37807141545067297376290837326, −6.15838809867181516406189013548, −5.86011195884456955968179936549, −5.21982033375412228088402830787, −4.15307300181887004694307978981, −3.39287719956716623984173546838, −2.89861087751840680271242942769, −1.66765527171259496989546726358, −0.72008104256037906846306939847, 0.72008104256037906846306939847, 1.66765527171259496989546726358, 2.89861087751840680271242942769, 3.39287719956716623984173546838, 4.15307300181887004694307978981, 5.21982033375412228088402830787, 5.86011195884456955968179936549, 6.15838809867181516406189013548, 7.37807141545067297376290837326, 7.76715879741981512004143848019

Graph of the ZZ-function along the critical line